Numerical methods in rock mechanics

Abstract
International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
CivilZone review paper
Numerical methods in rock mechanics $
L. Jing a, *, J.A. Hudson a,b
a Division of Engineering Geology, Royal Institute of Technology, S-100 44 Stockholm, Sweden
b Imperial College and Rock Engineering Consultants, 7 The Quadrangle, Welwyn Garden City, AL8 6SG, UK
Accepted 19 May 2002
The purpose of this CivilZone review paper is to present the techniques, advances, problems and likely future development
directions in numerical modelling for rock mechanics and rock engineering. Such modelling is essential for studying the fundamental
processes occurring in rock, for assessing the anticipated and actual performance of structures built on and in rock masses, and
hence for supporting rock engineering design. We begin by providing the rock engineering design backdrop to the review in Section
1. The states-of-the-art of different types of numerical methods are outlined in Section 2, with focus on representations of fractures
in the rock mass. In Section 3, the numerical methods for incorporating couplings between the thermal, hydraulic and mechanical
processes are described. In Section 4, inverse solution techniques are summarized. Finally, in Section 5, we list the issues of special
difficulty and importance in the subject. In the reference list, ‘significant’ references are asterisked and ‘very significant’ references
are doubly asterisked. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Review; Rock mechanics; Numerical modelling; Design; Coupled processes; Outstanding issues
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
2. Numerical methods for rock mechanics: states-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . 411
2.1. The FDM and related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
2.2. The FEM and related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
2.3. The BEM and related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
2.4. The distinct element method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
2.5. The DFN method related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
2.6. Hybrid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
2.7. Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
3. Coupled thermo-hydro-mechanical (THM) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
4. Inverse solution methods and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
4.1. Displacement-based back analysis for rock engineering . . . . . . . . . . . . . . . . . . . . . . . 417
4.2. Pressure-based inverse solution for groundwater flow and reservoir analysis . . . . . . . . . . . . 417
5. Conclusions and remaining issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
$ This paper was commissioned by Elsevier Science as part of its CivilZone initiative to generate review articles in civil engineering subjects.
*Corresponding author. Division of Engineering Geology, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Tel.: +46-8-790-6808;
fax.: +46-8-790-6810.
E-mail addresses: lanru@kth.se (L. Jing).
1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 1365-1609(02)00065-5

410
1. Introduction
L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
Because rock mechanics modelling has developed for
the design of rock engineering structures in different
circumstances and different purposes, and because
different modelling techniques have been developed,
we now have a wide spectrum of modelling and design
approaches. These approaches can be presented in
different ways. A categorization into eight approaches
based on four methods and two levels is illustrated in
Fig. 1.
The modelling and design work starts with the
objective, the top box in Fig. 1. Then there are the eight
modelling and design methods in the main central box.
The four columns represent the four main modelling
methods:
Method A— design based on previous experience.
Method B— design based on simplified models.
Method C— design based on modelling which attempts
to capture most relevant mechanisms.
Method D— design based on ‘all-encompassing’
modelling.
There are two rows in the large central box in Fig. 1.
The top row, Level 1, includes methods in which there is
an attempt to achieve one-to-one mechanism mapping
in the model. In other words, a mechanism which is
thought to be occurring in the rock reality and which
Site
Investigation
Objective
Method A Method B Method C Method D
Use of
pre-existing
standard
methods
Precedent type
analyses and
modifications
Analytical
methods,
stress-based
Rock mass
classification,
RMR, Q, GSI
Design based on forward analysis Design based on back analysis
Construction
is to be included in the model is modelled directly, such
as an explicit stress–strain relation. Conversely, the
lower row, Level 2, includes methods in which such
mechanism mapping is not totally direct, e.g. the use of
rock mass classification systems. Some of the rock mass
characterization parameters will be obtained from site
investigation, the left-hand box. Then the rock engineering
design and construction proceeds, with feedback
loops to the modelling from construction.
The review is directed at Methods C and D in the
Level 1 top row in the central box of Fig. 1. An
important point is that, in rock mechanics and
engineering design, having insufficient data is a way of
life, rather than a local difficulty—which is why the
empirical approaches, i.e. classification systems, have
been developed and are still required. So, we will also
be discussing the subjects of the representative elemental
volume (REV), homogenisation/upscaling, back analysis
and inverse solutions. These are fundamental
problems associated with modelling, and are relevant
to all the A, B, C & D method categories in Fig. 1.
The most commonly applied numerical methods for
rock mechanics problems are:
(1) Continuum methods—the finite difference method
(FDM), the finite element method (FEM), and the
boundary element method (BEM).
(2) Discrete methods—the discrete element method
(DEM), discrete fracture network (DFN) methods.
(3) Hybrid continuum/discrete methods.
Basic
numerical
methods, FEM,
BEM, DEM,
hybrid
Database
expert
systems, &
other systems
approaches
Extended
numerical
methods,
fully-coupled
models
Integrated
systems
approaches,
internet-based
Level 1
1:1 mapping
Level 2
Not 1:1 mapping
Fig. 1. The four basic methods, two levels, and hence eight different approaches to rock mechanics modelling and providing a predictive capability
for rock engineering design [1].

L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 411
The choice of continuum or discrete methods depends
on many problem-specific factors, and mainly on the
problem scale and fracture system geometry. The
continuum approach can be used if only a few fractures
are present and if fracture opening and complete block
detachment are not significant factors. The discrete
approach is most suitable for moderately fractured rock
masses where the number of fractures is too large for
the continuum-with-fracture-elements approach, or
where large-scale displacements of individual blocks
are possible. There are no absolute advantages of one
method over another. However, some of the disadvantages
of each type can be avoided by combined
continuum-discrete models, termed hybrid models.
In this review, we concentrate on the states-of-the-art
of the capability and utility of the numerical methods
for rock mechanics purposes and note the outstanding
issues to be solved. The emphasis is on civil engineering
applications, but the information applies to all branches
of rock engineering, and is supported here by a
moderately extensive literature reference source.
2. Numerical methods for rock mechanics: states-of-theart
2.1. The FDM and related methods
The basic technique in the FDM is the discretization
of the governing partial differential equations (PDEs) by
replacing the partial derivatives with differences defined
at neighbouring grid points. The grid system is a
convenient way of generating objective function values
at sampling points with small enough intervals between
them, so that errors thus introduced are small enough to
be acceptable. With proper formulations, such as static
or dynamic relaxation techniques, no global system of
equations in matrix form needs to be formed and solved.
The formation and solution of the equations are
localized, which is more efficient for memory and
storage handling in the computer implementation. No
local trial (or interpolation) functions are employed to
approximate the PDE in the neighbourhoods of the
sampling points, as is done in the FEM and BEM. It is
therefore the most direct and intuitive technique for the
solution of the PDEs. This also provides the additional
advantage of more straightforward simulation of complex
constitutive material behaviour, such as plasticity
and damage, without iterative solutions of predictorcorrector
mapping schemes which must be used in other
numerical methods using global matrix equation systems,
as in the FEM or BEM.
The conventional FDM with regular grid systems
does suffer from shortcomings, most of all in its
inflexibility in dealing with fractures, complex boundary
conditions and material heterogeneity. This makes the
standard FDM generally unsuitable for modelling
practical rock mechanics problems. However, significant
progress has been made in the FDM with irregular
meshes, such as triangular grid or Voronoi grid systems,
which leads to the so-called Control Volume, or Finite
Volume, techniques. Voronoi polygons grow from
points to fill the space, as opposed to tessellations where
the polygons are formed by lines cutting the plane, or by
building up a mosaic from pre-existing polygonal
shapes.
The FVM can be formulated with primary variables
(e.g. displacement) at the centres of cells (elements) or at
nodes (grid points) for an unstructured grid. It is also
possible to consider different material properties in
different cells. The FVM approach is therefore as
flexible as the FEM in handling material heterogeneity,
mesh generation, and treatment of boundary conditions
with unstructured grids of arbitrary shapes. It has
similarities with the FEM and is also regarded as a
bridge between the FDM and FEM. The original
concept and early code development in FVM for stress
analysis problems can be traced to the work in [2], using
a vertex-centred scheme with a quadrilateral grid. At
present, the most well-known computer code for stress
analysis for rock engineering problems using the FVM/
FDM approach is perhaps the FLAC code group [3],
with a vertex scheme of triangular and/or quadrilateral
grids.
Explicit representation of fractures is not easy in the
FDM/FVM because they require continuity of the
functions between the neighbouring grid points. In
addition, it is not possible to have special ‘fracture
elements’ in the FDM or FVM as in the FEM. In fact,
the inability to incorporate explicit representation of
fractures is the weak point of the FDM/FVM approach
for rock mechanics. On the other hand, the FDM/FVM
models have been used to study the mechanisms of
fracturing processes, such as shear-band formation in
the laboratory testing of rock and soil samples [4], and
fault system formation as a result of tectonic loading [5].
This is achieved via a process of material failure or
damage propagation at the grid points or cell centres,
without creating fracture surfaces in the models.
The FVM is one of the most popular numerical
methods in rock engineering, with applications covering
almost all aspects of rock mechanics, e.g. slope stability,
underground openings, coupled hydro-mechanical or
thermo-hydro-mechanical processes, rock mass characterization,
tectonic process, and glacial dynamics. The
most comprehensive coverage in this regard can be seen
in [6].
2.2. The FEM and related methods
The FEM is perhaps the most widely applied
numerical method across the science and engineering

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fields. Since its origin in the early 1960s, much FEM
development work has been specifically oriented towards
rock mechanics problems, as illustrated in [7–10].
This has been because it was the first numerical method
with enough flexibility for the treatment of material
heterogeneity, non-linear deformability (mainly plasticity),
complex boundary conditions, in situ stresses and
gravity. Also, the method appeared in the late 1960s and
early 1970s, when the traditional FDM with regular
grids could not satisfy these essential requirements for
rock mechanics problems. It out-performed the conventional
FDM because of these advantages.
Representation of rock fractures in the FEM has been
motivated by rock mechanics needs since the late 1960s,
with the most notable contributions reported in [11,12].
The well-known ‘Goodman joint element’ in rock
mechanics literature has been widely implemented in
FEM codes and applied to many practical rock
engineering problems. However, these models are
formulated based on continuum assumptions—so that
large-scale opening, sliding, and complete detachment of
elements are not permitted. The zero thickness of the
Goodman joint element causes numerical ill-conditioning
due to large aspect ratios (the ratio of length to
thickness) of joint elements, and was improved by joint
elements developed later in [13–20].
Despite these efforts, the treatment of fractures and
fracture growth remains the most important limiting
factor in the application of the FEM for rock mechanics
problems. The FEM suffers from the fact that the global
stiffness matrix tends to be ill-conditioned when many
fracture elements are incorporated. Block rotations,
complete detachment and large-scale fracture opening
cannot be treated because the general continuum
assumption in FEM formulations requires that fracture
elements cannot be torn apart.
When simulating the process of fracture growth, the
FEM is handicapped by the requirement of small
element size, continuous re-meshing with fracture
growth, and conformable fracture path and element
edges. This overall shortcoming makes the FEM less
efficient in dealing with fracture problems than its BEM
counterparts. However, special algorithms have been
developed in an attempt to overcome this disadvantage,
e.g. using discontinuous shape functions [21] for implicit
simulation of fracture initiation and growth through
bifurcation theory, the ‘enriched FEM’ and ‘generalized
FEM’ approaches [22–25]. The treatment of fractures is
at the element level: the surfaces of the fractures are
defined by assigned distance functions so that their
representation requires only nodal function values,
represented by an additional degree of freedom in the
trial functions, a jump function along the fracture and a
crack tip function at the tips. The motions of the
fractures are simulated using the level sets technique and
no pre-defined fracture elements are needed. In the
Generalized FEM, the meshes can be independent of the
problem geometry. In [26], the enriched FEM method
was applied to tunnel stability analysis with fractures
simulated as displacement discontinuities.
The Generalized FEM is in many ways similar to the
‘manifold method’ except for the treatment of fractures
and discrete blocks [27–29]. The manifold method
uses the truncated discontinuous shape functions to
simulate the fractures and treats the continuum bodies,
fractured bodies and assemblage of discrete blocks in a
unified form, and is a natural bridge between the
continuum and discrete representations. The method is
formulated using a node-based star covering system for
constructing the trial functions, where a node is
associated with a covering star, which can be a set of
standard FEM elements associated with the node or
generated using least-square kernel techniques with
general shapes. The integration, however, is performed
analytically using Simplex integration techniques. The
manifold method can also have meshes independent of
the domain geometry, and therefore the meshing task is
greatly simplified and simulation of the fracturing
process does not need remeshing. The technique has
been extended for applications to rock mechanics
problems with large deformations and crack propagation
[30,31]. Most of the publications are included in
the series of proceedings of the ICADD 1 conferences
[32–35].
The mesh generation, with complex interior structures
and exterior boundaries, is a demanding task when
applying the FEM to practical problems. The problem is
critical when dealing with 3-D problems with complex
geometry. Significant progress has been made in the last
decade in the ‘meshless’ (or ‘meshfree’, ‘element-free’)
method that simplifies greatly the meshing tasks. In this
approach, the trial functions are no longer standard, but
generated from neighbouring nodes within a domain of
influence by different approximations, such as the leastsquares
technique. A comprehensive overview of the
method is given in [36]. From the pure computing
performance point of view, it has not yet outperformed
the FEM techniques, but it has potential for civil
engineering problems in general, and rock mechanics
applications in particular—due to its flexibility in
treatment of fractures, as reported by Zhang et al. [37]
for analysis of jointed rock masses with block-interface
models, and by Belytschko et al. [38] for fracture growth
in concrete. A contact-detection algorithm using the
meshless technique was also reported by Li et al. [39]
that may pave the way for extending the meshless
technique to discrete block system modelling. The
concept was also extended to the BEM.
1
ICADD: International Conference on Analysis of Discontinuous
Deformations.

L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 413
2.3. The BEM and related methods
Unlike the FEM and FDM methods, the BEM
approach initially seeks a weak solution at the global
level through a numerical solution of an integral
equation derived using Betti’s reciprocal theorem and
Somigliana’s identity. The introduction of isoparametric
elements using different orders of shape functions, in the
same fashion as that in FEM, in [40,41], greatly
enhanced the BEMs applicability for stress analysis
problems.
The most notable original developments of the BEM
application in the field of rock mechanics may be
attributed to early works reported in [42–44] which was
quickly followed by many others, as reported in [45–48].
The applications were for general stress and deformation
analysis for underground excavations, soil-structure
interactions, groundwater flow and fracturing processes.
Notable examples of the work are as follows:
* Stress analysis of underground excavations with and
without fractures [49–55].
* Dynamic problems [56–58].
* Back analysis of in situ stress and elastic properties
[59,60].
* Borehole tests for permeability measurements [61].
Since the early 80s, an important developmental
thrust concerns BEM formulations for coupled thermo-mechanical
and hydro-mechanical processes, such
as the work reported in [62–64]. Due to the BEMs
advantage in reducing model dimensions, 3-D applications
are also reported, especially using the displacement
discontinuity method (DDM) for stress analysis, such as
in [65–67]. The DDM approach [68] is most suitable for
fracture growth simulations and was extended and
applied to rock fracture problems for two-dimensional
[69] and three-dimensional [70,71] problems. Its counterpart,
the fictitious stress method [44], is also an indirect
BEM approach suitable for stress simulations.
To simulate fracture growth using the standard BEM,
two techniques have been proposed. One is to divide the
problem domain into multiple sub-domains with fractures
along their interfaces, with a pre-assumed fracture
path, [72]; and the second is the dual boundary element
method (DBEM) using displacement and traction
boundary equations at opposite surfaces of fracture
elements [73,74]. However, the original concept of using
two independent boundary integral equations for
fracture analysis, one displacement equation and another
with its normal derivatives, was developed first in
[41]. Special crack tip elements, such as developed in
[75–76], should be used at the fracture tips to account
for the stress and displacement singularity.
A special formulation of the BEM, called the
Galerkin BEM, or GBEM, produces a symmetric
coefficient matrix by double integration of the tradi-
tional boundary integral twice multiplied by a weighted
trial function in the Galerkin sense of a weighted
residual formulation. A recent review of the GBEM is
given in [77] and an application for rock mechanics
problems was reported in [78].
Inclusion of source terms, such as body forces, heat
sources, sink/source terms in potential problems, etc.,
leads to domain integrals in the BEM. This problem will
also appear when considering initial stress/strain effects
and non-linear material behaviour, such as plastic
deformation. The traditional technique for dealing with
such domain integrals is the division of the domain into
a number of internal cells, which essentially eliminates
the advantages of the BEMs ‘boundary only’ discretization.
Different techniques have been developed over the
years to overcome this difficulty [79], most notably
the dual reciprocity method (DRM) [80]. In [81] the
approach is applied for solving groundwater flow
problems.
The main advantage of the BEM is the reduction of
the model dimension by one, with much simpler mesh
generation and therefore input data preparation, compared
with the FEM and FDM. In addition, solutions
inside the domain are continuous, unlike the point-wise
discontinuous solutions obtained using the FEM and
FDM. However, in general, the BEM is not as efficient
as the FEM in dealing with material heterogeneity—
because it cannot have as many sub-domains as
elements in the FEM. The BEM is also not as efficient
as the FEM in simulating non-linear material behaviour,
such as in plasticity and damage evolution processes
because domain integrals are often presented in these
problems. The BEM is more suitable for solving
problems of fracturing inhomogeneous and linearly
elastic bodies.
2.4. The distinct element method (DEM)
Rock mechanics is one of the disciplines from which
the concept of DEM originated [82–85]. The key
concepts of the DEM are that the domain of interest
is treated as an assemblage of rigid or deformable
blocks/particles and that the contacts among them need
to be identified and continuously updated during the
entire deformation/motion process, and be represented
by appropriate constitutive models. The theoretical
foundation of the method is the formulation and
solution of equations of motion of rigid and/or
deformable bodies using implicit (based on FEM
discretization) and explicit (using FDM/FVM discretization)
formulations. The method has a broad variety of
applications in rock mechanics, soil mechanics, structural
analysis, granular materials, material processing,
fluid mechanics, multibody systems, robot simulation,
computer animation, etc. It is one of most rapidly
developing areas of computational mechanics. The basic

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L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
difference between the DEM and continuum-based
methods is that the contact patterns between components
of the system are continuously changing with the
deformation process for the former, but are fixed for
the latter.
The most representative explicit DEM methods are
the computer codes UDEC and 3DEC for two and
three-dimensional problems in rock mechanics [86–92].
The hybrid technique with distinct element and boundary
element methods was also developed [93] to treat the
effects of the far-field most efficiently for two-dimensional
problems. Similar but different formulations and
numerical codes with the same principles have also been
developed and applied to various problems, such as rigid
block motions [94,95], plate bending [96], and the blockspring
model (BSM) [97–99] which is essentially a subset
of the explicit DEM by treating blocks as rigid bodies.
Due mainly to its conceptual attraction in the explicit
representation of fractures, the distinct element method
has been enjoying wide application in rock engineering.
A large quantity of associated publications has been
published, especially in conference proceedings. Some of
the publications are referenced here to show the wide
range of applicability of the methods.
* Underground works [100–109].
* Rock dynamics [110,111].
* Nuclear waste repository design and performance
assessment [112–114].
* Reservoir simulations [115].
* Fluid injection [116–118].
* Rock slopes [119].
* Laboratory test simulations and constitutive model
development [120–122].
* Stress-flow coupling [123,124].
* Hard rock reinforcement [125].
* Intraplate earthquakes [126].
* Well and borehole stability [127,128].
* Rock permeability characterization [129].
* Acoustic emission in rock [130].
* Derivation of equivalent properties of fractured rocks
[131,132].
A recent book [133] includes references to a collection of
explicit DEM application papers for various aspects of
rock engineering.
The seminal DEM work for granular materials for
geomechanics and civil engineering application is in a
series of papers [134–138]. The development and
applications are mostly reported in series of proceedings
of symposia and conferences, such as in [139–140] in the
field of geomechanics. The most well-known codes in
this field are the PFC codes for both two-dimensional
and three-dimensional problems [141], and the DMC
code [142,143]. The method has been widely applied to
many different fields such as soil mechanics, the
processing industry, non-metal material sciences and
defence research. More extensive coverage of this
subject will be given in a later expanded version of
this review.
The implicit DEM is represented mainly by the
discontinuous deformation analysis (DDA) approach,
originated in [144,145], and further developed in
[146,147] for stress-deformation analysis, and in
[148,149] for coupled stress-flow problems. Numerous
other extensions and improvements have been implemented
over the years in the late 90s, with the bulk of
the publications appearing in a series of ICADD
conferences [32–35]. The method uses standard FEM
meshes over blocks and the contacts are treated using
the penalty method. Similar approaches were also
developed in [150–152] using four-noded blocks as the
standard element, and was called the discrete finite
element method. Another similar development, called
the combined finite-discrete element method [153–155],
considers not only the block deformation but also
fracturing and fragmentation of the rocks. However, in
terms of development and application, the DDA
approach occupies the front position. DDA has two
advantages over the explicit DEM: relatively larger time
steps; and closed-form integrations for the stiffness
matrices of elements. An existing FEM code can also be
readily transformed into a DDA code while retaining all
the advantageous features of the FEM.
The DDA method has emerged as an attractive model
for geomechanical problems because its advantages
cannot be replaced by continuum-based methods or by
the explicit DEM formulations. It was also extended to
handle three-dimensional block system analysis [156]
and use of higher order elements [157], plus more
comprehensive representation of the fractures [158]. The
code development has reached a certain level of maturity
with applications focusing mainly on tunnelling, caverns,
fracturing and fragmentation processes of geological
and structural materials and earthquake effects
[159–165].
Similar to the DEM, but without considering block
deformation and motion, is the Key Block approach,
initiated independently by Warburton [166,167] and
Goodman and Shi [168], with a more rigorous
topological treatment of block system geometry in the
latter (see also [169,170]). This is a special method for
analysis of stability of rock structures dominated by the
geometrical characteristics of the rock blocks and hence
the fracture systems. It does not utilize any stress and
deformation analysis, but identifies the ‘key blocks’ or
‘keystones’, which are formed by intersecting fractures
and excavated free surfaces in the rock mass which have
the potential for sliding and rotation in certain directions.
Key block theory, or simply block theory, and the
associated code development enjoy wide applications in
rock engineering, with further development considering
Monte Carlo simulations and probabilistic predictions

L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 415
[171–175], water effects [176], linear programming [177],
finite block size effect [178] and secondary blocks [179].
Predictably, the major applications are in the field of
tunnel and slope stability analysis, such as reported in
[180–186].
2.5. The DFN method related methods
The DFN method is a special discrete model that
considers fluid flow and transport processes in fractured
rock masses through a system of connected fractures.
The technique was created in the early 1980s for
both two-dimensional and three-dimensional problems
[187–197], and is most useful for the study of flow in
fractured media in which an equivalent continuum
model is difficult to establish, and for the derivation of
equivalent continuum flow and transport properties of
fractured rocks [198,199]. A large number of publications
have been reported, and systematic presentation
and evaluation of the method have also appeared in
books [200–203]. The method enjoys wide applications
for problems of fractured rocks, perhaps mainly due to
its conceptual attractiveness. There are many different
DFN formulations and computer codes, but most
notable are the approaches and codes FRACMAN/
MAFIC [204] and NAPSAC [205–207] with many
applications for rock engineering projects over the years.
The stochastic simulation of fracture systems is the
geometric basis of the DFN approach and plays a
crucial role in the performance and reliability of the
fracture system model in the same way as for the DEM
[208,209]. A critical issue is the treatment of bias in
estimation of the fracture densities, trace lengths and
connectivity from conventional surface or borehole
mappings. Recent development using circular windows
is reported in [210,211].
Solution of flow fields for individual fracture disks in
the DFN uses closed-form solutions, the FEM and
BEM mesh, pipe models and channel lattice models.
Closed-form solutions exist, at present, only for planar
fractures with parallel surfaces of regular (i.e. circular or
rectangular discs) shapes [212,213]. For fractures with
general shapes, the FEM discretization technique is
perhaps the most well-known techniques used in the
DFN codes FRACMAN/MAFIC and NAPSAC. The
use of BEM discretization is reported in [194,195].
The pipe model [214] and the channel lattice model
[215] provide simpler representations of the fracture
system geometry, the latter being more suitable for
simulating the complex flow behaviour inside the
fractures, such as the ‘channel flow’ phenomenon.
Computationally, they are less demanding than the
FEM and BEM models because the solutions for the
flow fields through the channel elements are analytical.
The fractal concept has also been applied to the DFN
approach in order to consider the scale dependence of
the fracture system geometry and for up-scaling the
permeability properties, using usually the full box
dimensions or the Cantor dust model [216–218]. Power
law relations have been also found to exist for
tracelengths of fractures and have been applied for
representing fracture system connectivity [219]. The
fracture–matrix interaction for DFN models was
reported in [220] using full FEM discretization and in
[221] with a simplified technique using a probabilistic
particle tracking technique. The effects of in situ stresses
on fracture aperture variations in DFN models were
reported in [222]. Below are some examples of developments
and applications of the DFN approach.
* Developments for multiphase fluid flow [223].
* Hot-dry-rock reservoir simulations [224–226].
* Characterization of permeability of fractured rocks
[227–234].
* Water effects on underground excavations and rock
slopes [235–237].
Despite the advantages of the DEM and DFN, lack of
knowledge of the geometry of the rock fractures limits
their more general application. In general, the detailed
geometry of fracture systems in rock masses cannot be
known and can only be roughly estimated. The adequacy
of the DEM and DFN are therefore highly dependent on
the interpretation of the in situ fracture system
geometry—which cannot be even moderately validated
in practice. The same problem applies to the continuumbased
models as well, but the requirement for explicit
fracture geometry representation in the DEM and DFN
models makes the drawback more acute, even with
multiple stochastic fracture system realizations. The
understanding and quantification of the system uncertainty
become more necessary in discrete models.
There are other numerical techniques such as percolation
theory and lattice models that are closely related to
DEM and DFN approaches. Reviews on their fundamentals
and applications are covered in the extended
version of this paper to be published later.
2.6. Hybrid models
Hybrid models are frequently used in rock engineering,
basically for flow and stress/deformation problems
of fractured rocks. The main types of hybrid models are
the hybrid BEM/FEM, DEM/FEM and DEM/BEM
models. The BEM is most commonly used for simulating
far-field rocks as an equivalent elastic continuum,
and the FEM and DEM for the non-linear or fractured
near-fields where explicit representations of fractures
and/or non-linear mechanical behaviour, such as plasticity,
are needed. This harmonizes the geometry of
the required problem resolution with the numerical
techniques available, thus providing an effective representation
of the far-field to the near-field rock mass.

416
The hybrid FEM/BEM was first proposed in [238],
then followed in [239,240] as a general stress analysis
technique. In rock mechanics, it has been used mainly
for simulating the mechanical behaviour of underground
excavations, as reported in [241–245]. The
coupling algorithms are also presented in detail in [48].
The hybrid DEM/BEM model was implemented
only for the explicit distinct element method, in the
code group of UDEC and 3DEC. The technique
was created in [246–248] for stress/deformation analysis.
In [249–250] a development of hybrid discrete-continuum
models was reported for coupled hydro-mechanical
analysis of fractured rocks, using combinations
of DEM, DFN and BEM approaches. In [251], a
hybrid DEM/FEM model was described, in which
the DEM region consists of rigid blocks and the
FEM region can have non-linear material behaviour.
A hybrid beam-BEM model was reported in [252]
to simulate the support behaviour of underground
openings, using the same principle as the hybrid BEM/
FEM model. In [253], a hybrid BEM-characteristics
method is described for non-linear analysis of rock
caverns.
The hybrid models have many advantages, but special
attention needs to be paid to the continuity or
compatibility conditions at the interfaces between
regions of different models, particularly when different
material assumptions are involved, such as rigid and
deformable block–region interfaces.
2.7. Neural networks
L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
All the numerical modelling methods described so far
are in the category of ‘1:1 mapping’, i.e., the Level 1
methods in Fig. 1. The neural network approach is a
‘non-1:1 mapping’ in the Methods C and D categories of
the Level 2 methods indicated in Fig. 1. The rock mass is
represented indirectly by a system of connected nodes,
but there is not necessarily any physical interpretation of
the geometrical or mechanistic location of the network’s
internal nodes, nor of their input and output values.
Such a ‘non-1:1 mapping’ system has its advantages and
disadvantages.
The advantages of neural networks are that
(1) The geometrical and physical constraints of the
problem, which dominate the governing equations
and constitutive laws when the 1:1 mapping
techniques are used, are not such a problem.
(2) Different kinds of neural networks can be applied
to a problem.
(3) There is the possibility that the ‘perception’ we
enjoy with the human brain may be mimicked in the
neural network, so that the programs can incorporate
judgements based on empirical methods and
experiences.
The disadvantages are that
(1) The procedure may be regarded as simply supercomplicated
curve fitting—because the program has
to be ‘taught’.
(2) The model cannot reliably estimate outside its range
of training parameters.
(3) Critical mechanisms might be omitted in the model
training.
(4) There is a lack of any theoretical basis for
verification and validation of the techniques and
their outcomes.
Neural network models provide descriptive and
predictive capabilities and, for this reason, have been
applied through the range of rock parameter identification
and engineering activities. Recent published works
on the application of neural networks to rock mechanics
and rock engineering includes the following publications.
* Stress–strain curves for intact rock [254].
* Intact rock strength [255,256].
* Fracture aperture [257].
* Shear behaviour of fractures [258].
* Rock fracture analysis [259].
* Rock mass properties [260,261].
* Microfracturing process in rock [262].
* Rock mass classification [263,264].
* Displacements of rock slopes [265].
* Tunnel boring machine performance [266].
* Displacements and failure in tunnels [267,268].
* Tunnel support [269,270].
* Surface settlement due to tunnelling [271].
* Earthquake information analysis [272].
* Rock engineering systems (RES) modelling [273,274].
* Rock engineering [275].
* Overview of the subject [276].
As evidenced by the list of highlighted references
above, the neural network modelling approach has been
widely applied and is considered to have significant
potential—because of its ‘non-1:1 mapping’ character
and because it may be possible in the future for such
networks to include creative ability, perception and
judgement, and be linked to the Internet. However, the
method has not yet provided an alternative to conventional
modelling, and it may be some time before it can
be used in the comprehensive Box 2D mode envisaged in
Fig. 1 and described in [277].
3. Coupled thermo-hydro-mechanical (THM) models
The couplings between the processes of heat transfer
(T), fluid flow (H) and stress/deformation (M) in
fractured rocks have become an increasingly important
subject since the early 1980s [278,279], mainly due to the

L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 417
modelling requirements for the design and performance
assessment of underground radioactive waste repositories,
and other engineering fields in which heat and
water play important roles, such as gas/oil recovery,
hot-dry-rock thermal energy extraction, contaminant
transport analysis and environment impact evaluation in
general. The term ‘coupled processes’ implies that the
rock mass response to natural or man-made perturbations,
such as the construction and operation of a
nuclear waste repository, cannot be predicted with
confidence by considering each process independently.
The THM coupling models are based on heat and
multiphase fluid flow in deformable and fractured porous
media, and have been mainly developed according to two
basic ‘partial’ but well established coupling mechanisms:
the thermo-elasticity theory of solids and the poroelasticity
theory developed by Biot, based on Hooke’s law of
elasticity, Darcy’s law of flow in porous media, and
Fourier’s law of heat conduction. The effects of the THM
couplings are formulated as three inter-related PDEs,
expressing the conservation of mass, energy and momentum,
for describing fluid flow, heat transfer and
deformation. The solution technique can be based on
continuum representations using mainly the FEM [280–
287], FVM [3] and the discrete approach using the
UDEC code without matrix flow but with heat convection
along fractures [288]. In [289] and [290] systematic
development of the governing equations and FEM
solution techniques are presented for porous continua.
Comprehensive studies, using both continuum and
discrete approaches, have been conducted in the
international DECOVALEX 2 projects for coupled
THM processes in fractured rocks and buffer materials
for underground radioactive waste disposal since 1992,
with results published in a series of reports [291,292], an
edited book [293] and two special issues of the
International Journal of Rock Mechanics and Mining
Sciences in [294,295]. Other applications are reported, as
examples, in the following list.
* Reservoir simulations [296–298].
* Partially saturated porous materials [299].
* Advanced numerical solution techniques for coupled
THM models [300–302].
* Soil mechanics [303].
* Simulation of expansive clays [304].
* Flow and mechanics of fractures [305].
* Nuclear waste repositories [306,307].
* Non-Darcy flow in coupled THM processes [308].
* Double-porosity model of porous media [309].
* Parallel formulations of coupled models for porous
media [310].
* Tunneling in cold regions [311].
2
DECOVALEX—DEvelopment of COupled Models and their
VAlidation against EXperiments.
The coupled processes and models represent a great
advance in rock mechanics towards an well founded
branch of physics. Their extensions to include chemical,
biochemical, electrical, acoustic and magnetic processes
have also started to appear in the literature and are an
indication of future research directions.
4. Inverse solution methods and applications
A large and important class of numerical methods in
rock mechanics and civil engineering practice is the
inverse solution techniques. The essence of the inverse
solution approach is to derive unknown material
properties, system geometry, and boundary or initial
conditions, based on a limited number of laboratory or
usually in situ measured values of some key variables,
using either least square or mathematical programming
techniques of error minimization. In the case of rock
engineering, the most widely applied inverse solution
technique is back analysis using measured displacements
in the field [312,313] and the inverse solution of rock
permeability using field pressure data.
4.1. Displacement-based back analysis for rock
engineering
Since displacements measured by extensometers with
multiple anchors and the convergence of tunnel walls are
the most directly measurable quantities in situ, and are
one of the most used primary variables in many
numerical methods, they have been used extensively to
derive rock properties over the years. The majority of
applications concern identification of constitutive properties
and parameters of rocks, using displacement
measurements from tunnels or slopes. Below are some
typical and recent examples of such applications.
* Underground excavations [314–325].
* Slopes and foundations [326–330].
* Initial stress field [331].
* Time-dependent rock behaviour [332,333].
* Consolidation [334].
4.2. Pressure-based inverse solution for groundwater flow
and reservoir analysis
The inverse solution has long been used in hydrogeology,
reservoir engineering (oil, gas and hot-dry-rock
(HDR) geothermal reservoirs) and geotechnical engineering
analyses of environmental impacts—as a critical
technique for estimating hydraulic properties of largescale
geological formations, such as permeability,
porosity, storativity, etc. by assuming hydraulic constitutive
laws of porous media based on Darcy’s law or
non-Newtonian fluid models. Complexity is increased

418
L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
when thermal processes are also involved, due to phase
changes during multiple-phased flow of fluids with
various states of saturation.
A comprehensive review of the subject is given in [335]
with the history of inverse solution development and
methods for the past 40 years, especially the application
of the stochastic approach using geostatistics. Some of
the most recent developments in this subject are
referenced below to illustrate the advances.
* Capillary pressure-saturation and permeability function
of two-phase fluids in soil [336].
* Water capacity of porous media [337].
* Unsaturated properties [338].
* Transmissivity, hydraulic head and velocity fields
[339].
* Hydraulic function estimation using evapotranspiration
fluxes [340].
* Use of BEM for inverse solution [341].
* Integral formulation [342].
* Hydraulic conductivity of rocks using pump test
results [343].
* Least-square penalty technique for steady-state aquifer
models [344].
* Maximum likelihood estimation method [345].
* Inversion using transient outflow methods [346].
* Use of geostatistics for transmissivity estimation
[347–350].
* Successive forward perturbation method [351].
A distinct advantage of the inverse solution technique
is the fact that the measured values in the field represent
the behaviour of a large volume of rock and the scaleeffects
of the constitutive parameters are automatically
included in the identified parameter values. It also
indicates a promising method for validation of constitutive
models and properties using back analysis with
field measurements. On the other hand, the uniqueness
of the solution is not guaranteed because the minimization
of the same objective error function may be
achieved through different paths.
5. Conclusions and remaining issues
Over the last three decades, advances in the use of
computational methods in rock mechanics have been
impressive—especially in specific numerical methods,
based on both continuum and discrete approaches, for
representation of fracture systems, for comprehensive
constitutive models of fractures and interfaces, and in
the development of coupled THM models. Despite
all the advances, our computer methods and codes can
still be inadequate when facing the challenge of
some practical problems, and especially when adequate
representation of rock fracture systems and fracture
behaviour are a pre-condition for successful modelling.
Issues of special difficulty and importance are the
following.
* Systematic evaluation of geological and engineering
uncertainties.
* Understanding and mathematical representation of
large rock fractures.
* Quantification of fracture shape, size, connectivity
and effect of fracture intersections for DFN and
DEM models.
* Representation of rock mass properties and behaviour
as an equivalent continuum and existence of the
REV.
* Representation of interface behaviour.
* Scale effects, homogenization and upscaling methods.
* Numerical representation of engineering processes,
such as excavation sequence, grouting and reinforcement.
* Time effects.
* Large-scale computational capacities.
The ‘model’ and the ‘computer’ are now integral
components of rock mechanics and rock engineering
studies. Indeed, numerical methods and computing
techniques have become daily tools for formulating
conceptual models and mathematical theories
integrating diverse information about geology, physics,
construction techniques, economy, the environment and
their interactions. This achievement has greatly enhanced
the development of modern rock mechanics—
from the traditional ‘empirical’ art of rock deformability
and strength estimation and support design, to the
rationalism of modern mechanics, governed by and
established on the three basic principles of physics:
mass, momentum and energy conservation.
As a result of numerical modelling experience over the
past decades, it has become abundantly clear that the
most important step in numerical modelling is, perhaps
counter-intuitively, not operating the computer code,
but the earlier ‘conceptualization’ of the problem in
terms of the dominant processes, properties, parameters
and perturbations, and their mathematical presentations.
The associated modelling steps of addressing the
uncertainties and estimating their significance via
sensitivity analyses is similarly important. Moreover,
success in numerical modelling for rock mechanics and
rock engineering can depend almost entirely on the
quality of the characterization of the fracture system
geometry, the physical behaviour of the individual
fractures and the interaction between intersecting
fractures. Today’s numerical modelling capability can
handle very large scale and complex equations systems,
but the quantitative representation of the physics of
fractured rocks remains generally questionable,
although much progress has been made in this direction.
The engineer must have a predictive capability for
design, and that predictive capability can only be

achieved if the key features of the rock reality have
indeed been captured in the model. Furthermore, the
engineer needs some reassurance that this is indeed
the case, which is why the authors place importance on
the concept of auditing rock mechanics modelling and
rock engineering design to ensure that the modelling is
adequate in terms of the modelling or engineering design
objective.
Acknowledgements
The authors would like to express their sincere
appreciation and gratitude to Professor B.H.G. Brady,
Professor Y. Ohnishi, Professor W.G. Pariseau and Dr
R.W. Zimmerman for their comments, suggestions,
corrections, and especially encouragement, in their
reviews of the extended version of this paper.
Introduction to references
In CivilZone review papers, the ‘significant’ and ‘very
significant’ references quoted in the review are highlighted,
as indicated here by the symbols * and **,
respectively. The asterisked references represent groundbreaking
developments or major advances in the subject,
or contain comprehensive review material.
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